 Quasistatic temperature of a conductively cooled film

For a film with fixed lower temperature $T_0$ and upper condition $-\kappa T_z=\alpha (T_s-T_0+\Delta T)$, negligible horizontal thermal diffusion and thermal advection reduce the heat equation to $T_{zz}=0$. Thus $T=T_0-\alpha \Delta Tz/(\kappa +\alpha h)$. The local approximation requires $(h/L{)}^2\ll 1$ and $U{h}^2/(\kappa L)\ll 1$, together with slow time variation relative to $h^2/\kappa$. For $\alpha h/\kappa \ll 1$, $T_s\simeq T_0-\Delta T\alpha h/\kappa$. The coefficient $\kappa$ is thermal diffusivity, and the boundary coefficient $\alpha$ is normalized accordingly.